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There's a few companies that make forstner bits that size. They're a bit pricey, but I've just found some on eBay UK at 90mm, by Milwaukee, for less than £37GBP. Should easily be able to pick one up over there for less than 50 bucks.

If you don't want to buy a bit, then the only other way I could think of doing it would be with a holesaw - cut 1" deep, chisel out the waste, drill another 1", chisel out the waste, and so on. The length of the holesaw and the chuck/front of the drill would fit within that diameter, so you should be able to get to 6" deep that way. It would be very time-consuming though, and the finish inside the hole wouldn't be great (don't know if that'd be a problem for you or not).

But like Ed said, if your stock would fit on a lathe (and you have access to one) that might be the easiest way.

I made about 20 log vases for a ladies charity. One of the ladies had a tree cut down on her property and had the 20 sections cut from limbs. The sections were about 8" in diameter and about 12" long. I don't remember what type of tree, but the wood was hard although very green.

I mounted the log sections on a lathe faceplate getting them as centered and trued as possible.

I started the hole using a 1 1/2" Forstner drill bit.

From there, I used bowl cutters to hollow out the hole for the vases. The vases were 3" in diameter and about 5" deep.

I had to keep the speed down so as to avoid "throwing" the log section. Very tricky.

I've done the hole saw thing in fir before. I drill down to the full depth of the hole saw, 1 1/2" - 1 3/4" depending upon how much of the arbor protudes inside the hole saw, take a 1/4" stainless rod, place it in the pilot hole that was drilled and then force it from side-to-side until the hole saw plug breaks off. If it's particularly stubborn (or if it's too green), I'll take a long taper punch, place it in the pilot hole and hammer it into the plug, a miniature version of log splitting if you will. You'll always be left with some vestige of the pilot hole remaining, so just go back at it until you get the depth you need. The only downside is that the interior walls will be quite rough and you'll probably want to get a reasonably flat bottom on the hole.

Three things from the video:1. The amount of different materials it can cut through because of the carbide teeth. 2. The additional cut depth of the hole saw.3. The ease of mounting the hole saw on the hole saw shaft.

I also liked the mounting of 2 different hole saws on the same shaft for increasing the size of the original hole. Starrett offers an "oops" adapter that works in the same manner, unfortunately it only works with a lot smaller diameter series of hole saws.